Solve for $x$ : $ 6|x - 10| - 1 = 5|x - 10| + 7 $
Answer: Subtract $ {5|x - 10|} $ from both sides: $ \begin{eqnarray} 6|x - 10| - 1 &=& 5|x - 10| + 7 \\ \\ { - 5|x - 10|} && { - 5|x - 10|} \\ \\ 1|x - 10| - 1 &=& 7 \end{eqnarray} $ Add ${1}$ to both sides: $ \begin{eqnarray} 1|x - 10| - 1 &=& 7 \\ \\ { + 1} &=& { + 1} \\ \\ 1|x - 10| &=& 8 \end{eqnarray} $ Simplify: $ |x - 10| = 8$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x - 10 = -8 $ or $ x - 10 = 8 $ Solve for the solution where $x - 10$ is negative: $ x - 10 = -8 $ Add ${10}$ to both sides: $ \begin{eqnarray} x - 10 &=& -8 \\ \\ {+ 10} && {+ 10} \\ \\ x &=& -8 + 10 \end{eqnarray} $ $ x = 2 $ Then calculate the solution where $x - 10$ is positive: $ x - 10 = 8 $ Add ${10}$ to both sides: $ \begin{eqnarray} x - 10 &=& 8 \\ \\ {+ 10} && {+ 10} \\ \\ x &=& 8 + 10 \end{eqnarray} $ $ x = 18 $ Thus, the correct answer is $x = 2 $ or $x = 18 $.